PHY 3221 Classical Mechanics, Homework #4, due 8:30 am, 02/20/12
1 This question concerns the line integral of work done along a specific path.
a) Consider a force F = (x2 + y 2 , 2xy) acting in the xy-plane. Calculate the work done in moving
along the circular arc from the point (1, 0) to the point (0, 1).
b) Consider a pendulum hanging under gravity. Calculate the work done in raising the pendulum bob
from its vertical equilibrium position along a circular arc until the pendulum string is horizontal.
c) A force is given by (yz, zx, xy) and a curve is parameterized by x = −u, y = 1 + u and z = u + u2 .
Calculate the work done along the curve, from the point (1, 0, 0) to the point (0, 1, 0).
1
2 Calculate the gradient of the following functions.
a) fa (x, y, z) = x2 + y 3 + z 4 .
√
x2 + y 2 + z 2 .
√
c) fc (x, y, z) = exp r, where r = x2 + y 2 + z 2 .
b) fb (x, y, z) = ln r, where r =
d) fd (x, y, z) = x4 + y 4 + z 4 + x2 y 2 + y 2 z 2 + z 2 x2 .
e) For which (if any) of these does the negative of the gradient give a conservative force?
2
3 A massless spring of spring constant k hangs vertically. When a mass m is attached to the lower
end, the spring stretches by an amount x0 .
a) Find x0 .
b) Assume that the zero of potential energy is at x0 . The mass is then disturbed by a vertical
impulse. Find the total potential energy of the system in terms of the displacement of the mass
from x0 .
c) If the mass is pulled a distance y from its equilibrium at x0 , find its subsequent maximum velocity.
3
4 A body is composed of a cylinder of radius R and length L and a hemisphere also of radius R.
Assume both are of the same density. The circular cross-section of the hemisphere is attached
to one circular end of the cylinder. The body is balanced on the hemisphere with the cylinder
section being vertical.
a) Find the center of mass of the whole system.
b) For what value of L is the center of mass at the center of the hemisphere (i.e., at the cylinderhemisphere interface).
c) If L is larger than this, what will happen to the body if it is tipped slightly away from equilibrium?
d) Explain why this does not happen if L is smaller than the value you found in part (b).
4